Number and ratio · GCSE Maths
Direct proportion
GCSE Maths direct proportion: y = kx, graphs through the origin, and unitary method for recipes, maps and currency when quantities rise together.
Direct proportion: as one doubles, the other doubles. y = kx — multiply by the constant, or find the value of one unit first.
The important bits
What you need to know
- 1
Two quantities are in direct proportion if they increase and decrease together at a constant ratio. y ∝ x means y = kx for some constant k.
- 2
The graph of y against x is a straight line through the origin (0, 0). The gradient k is the constant of proportionality.
- 3
To find k, divide y by x: if 5 books cost £12, k = 12/5 = £2.40 per book. Then 8 books cost 8 × 2.40 = £19.20.
- 4
The unitary method finds the value of one unit first, then scales: 3 pens for £4.50 → one pen £1.50 → 7 pens £10.50.
- 5
Direct proportion appears in maps (scale factor), recipes (same multiplier on every ingredient), and exchange rates (more pounds buys more euros).
- 6
Equations such as y = 3x or C = 5n are direct proportion. y = 3x + 2 is not — the +2 shifts the line off the origin.
- 7
If y is proportional to x², write y = kx². Doubling x quadruples y. Read the power carefully: y ∝ x² is not linear on a y–x graph.
- 8
Check proportionality by dividing: if y/x is always the same constant for every pair, the relationship is direct proportion.
Quotations worth analysing
Short evidence. Real method.
“y = kx”
k is the constant of proportionality. Students who add a fixed amount (y = kx + c) have described a line not through the origin — not direct proportion.
“Find the value of one unit, then multiply”
This works for direct proportion and is harder to reverse than a scale factor. Still write the one-unit line before you scale.
“Graph through the origin”
A straight line with positive gradient through (0, 0) means direct proportion. A line that misses the origin has a fixed charge plus proportion.
Go deeper
Constant k from a table
A car travels 120 km in 2 hours, 180 km in 3 hours, 240 km in 4 hours. Distance ÷ time is always 60, so speed = 60 km/h and distance = 60t. That is direct proportion: double the time, double the distance. If a question says “distance is proportional to time”, write d = kt, substitute one pair to find k, then answer the rest. Tables that do not give a constant ratio are not direct proportion — check every row. Recipe: 200 g flour for 8 buns. Flour f = 25g per bun. For 14 buns: 14 × 25 = 350 g. Every ingredient must use the same bun multiplier 14/8 = 1.75, or the unitary 25 g per bun. Mixing “add 50 g because we added 6 buns” on one ingredient only ruins the recipe.
Go deeper
y = kx² and other powers
If y is proportional to x², y = kx². When x = 3, y = 18, so k = 18/9 = 2 and y = 2x². When x = 5, y = 50, not 10. The exam phrase is “y is proportional to the square of x”. Graph y against x² to get a straight line through the origin with gradient k. Similarly y ∝ 1/x is inverse, not direct. Direct proportion questions on Foundation often stay with y = kx. Higher may use squares or cubes. Always write the proportional relationship in words as an equation before substituting numbers.
Go deeper
Maps, scale and currency
A map scale 1 : 25 000 means 1 cm on the map represents 25 000 cm (250 m) in real life. 4 cm on the map is 4 × 25 000 cm = 100 000 cm = 1 km real distance. That is direct proportion between map length and real length. Currency: £1 = €1.15 means pounds and euros are directly proportional at rate 1.15 euros per pound. £40 = €46. To find pounds for €69: one euro costs 1/1.15 pounds, so €69 = 69/1.15 = £60. Write the unit rate and the direction (pounds per euro or euros per pound) before multiplying.
See the idea in action
The cost C pounds is directly proportional to the number of hours h. 5 hours cost £35. Find the cost of 12 hours and the value of h when C = £84. Step 1: C = kh. 35 = 5k, so k = 7. Equation: C = 7h. Step 2: 12 hours: C = 7 × 12 = £84. Step 3: C = 84 when 7h = 84, so h = 12 hours. Check: gradient £7 per hour, graph through origin.
Exam technique
Turn knowledge into marks
Write “C = kh” or “y = kx” before substituting. The constant k is a method mark. For recipes, multiply every ingredient by the same scale factor.
Common mistakes
Do not give these marks away
- 01
Using y = kx + c when the relationship is direct proportion (line must pass through the origin).
- 02
Adding a fixed amount per extra item instead of multiplying all ingredients by the same factor.
- 03
Treating y ∝ x² as y = kx and using the wrong power.
y is directly proportional to x. When x = 4, y = 20. When x = 10, y is
A50
B24
C40
D5
Show the answer
50. k = 20/4 = 5, so y = 5x. When x = 10, y = 50. 24 would be adding 4 each time. 5 is k itself, not y.
Quick questions
If this is the bit you searched
How do I know if a relationship is direct proportion?
The ratio y/x is constant, and the graph of y against x is a straight line through the origin. y = 3x + 1 fails the origin test.
What is the constant of proportionality?
The number k in y = kx. It is the gradient of the graph and the value of y when x = 1 in simple cases.
Is a straight line always direct proportion?
Only if it goes through (0, 0). A line with equation y = 2x + 5 has gradient 2 but is not direct proportion because of the +5.
How does direct proportion differ from ratio sharing?
Sharing splits a fixed total into parts. Direct proportion links two variables that grow together — more hours, more cost, with no fixed total.